Proposition 4.8.6.24. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. The following conditions are equivalent:
The functor $F$ is $(n+1)$-faithful.
The comparison map $F': \operatorname{\mathcal{C}}\rightarrow \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ of Remark 4.8.6.18 is an equivalence of $\infty $-categories.
Proof. It follows from Proposition 4.8.6.17 (and Proposition 4.8.6.12) that the comparison map $G: \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \rightarrow \operatorname{\mathcal{D}}$ is $(n+1)$-faithful. By virtue of Remark 4.8.3.29, we can replace $(1)$ by the following condition:
The functor $F'$ is $(n+1)$-faithful: that is, it is $m$-full for $m \geq n+2$.
Since $F'$ is also $m$-full for $m \leq n+1$ (Corollary 4.8.6.19), the equivalence $(1') \Leftrightarrow (2)$ follows from Theorem 4.8.4.1. $\square$